Find the value of k for which the quadratic equation 4x2 - 2 (k + 1)

1.	Find the value of k for which the quadratic equation 4x2 - 2 (k + 1) x + (k + 4) = 0 has equal roots.
Answer» Since roots are equal ∴ d = 0 ….(1) 4x² - 2 (k + 1) x + (k + 4) = 0 d = b² – 4ac d = (–2 (k + 1)² – 4 (4) (k + 4) d = (– 2k – 2)² – 16k – 64 d = 4k² + 4 + 8k – 16k – 64 (∵ (a – b)² = a² + b² – 2ab) d = 4k² – 8k – 60 From (1), d = 0 ∴ Equation will be: 4k² – 8k – 60 = 0 Dividing by 4 k² – 2k – 15 = 0 4k² – 5k + 3k – 15 = 0 k (k – 5) + 3(k – 5) = 0 (k – 5) (k + 3) = 0 K – 5 = 0 k + 3 = 0 K = 5 k = –3

Find the value of k for which the quadratic equation 4x2 - 2 (k + 1) x + (k + 4) = 0 has equal roots.

Answer»

Since roots are equal

∴ d = 0 ….(1)

4x² - 2 (k + 1) x + (k + 4) = 0

d = b² – 4ac

d = (–2 (k + 1)² – 4 (4) (k + 4)

d = (– 2k – 2)² – 16k – 64

d = 4k² + 4 + 8k – 16k – 64

(∵ (a – b)² = a² + b² – 2ab)

d = 4k² – 8k – 60

From (1), d = 0

∴ Equation will be:

4k² – 8k – 60 = 0

Dividing by 4

k² – 2k – 15 = 0

4k² – 5k + 3k – 15 = 0

k (k – 5) + 3(k – 5) = 0

(k – 5) (k + 3) = 0

K – 5 = 0 k + 3 = 0

K = 5 k = –3